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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ofcc | Structured version Visualization version GIF version |
Description: Left operation by a constant on a mixed operation with a constant. (Contributed by Thierry Arnoux, 31-Jan-2017.) |
Ref | Expression |
---|---|
ofcc.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
ofcc.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
ofcc.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
Ref | Expression |
---|---|
ofcc | ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ofcc.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
2 | fnconstg 6776 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → (𝐴 × {𝐵}) Fn 𝐴) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 × {𝐵}) Fn 𝐴) |
4 | ofcc.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | ofcc.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
6 | fvconst2g 7199 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵) | |
7 | 1, 6 | sylan 580 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵) |
8 | 3, 4, 5, 7 | ofcfval 33084 | . 2 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) |
9 | fconstmpt 5736 | . 2 ⊢ (𝐴 × {(𝐵𝑅𝐶)}) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)) | |
10 | 8, 9 | eqtr4di 2790 | 1 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {csn 4627 ↦ cmpt 5230 × cxp 5673 Fn wfn 6535 ‘cfv 6540 (class class class)co 7405 ∘f/c cofc 33081 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-ofc 33082 |
This theorem is referenced by: (None) |
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